3.8.41 \(\int \frac {a+b x}{\sqrt {c x^2}} \, dx\)

Optimal. Leaf size=29 \[ \frac {a x \log (x)}{\sqrt {c x^2}}+\frac {b x^2}{\sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \begin {gather*} \frac {a x \log (x)}{\sqrt {c x^2}}+\frac {b x^2}{\sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/Sqrt[c*x^2],x]

[Out]

(b*x^2)/Sqrt[c*x^2] + (a*x*Log[x])/Sqrt[c*x^2]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a+b x}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (b+\frac {a}{x}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {b x^2}{\sqrt {c x^2}}+\frac {a x \log (x)}{\sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.66 \begin {gather*} \frac {x (a \log (x)+b x)}{\sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/Sqrt[c*x^2],x]

[Out]

(x*(b*x + a*Log[x]))/Sqrt[c*x^2]

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IntegrateAlgebraic [A]  time = 0.02, size = 26, normalized size = 0.90 \begin {gather*} \sqrt {c x^2} \left (\frac {a \log (x)}{c x}+\frac {b}{c}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)/Sqrt[c*x^2],x]

[Out]

Sqrt[c*x^2]*(b/c + (a*Log[x])/(c*x))

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fricas [A]  time = 1.02, size = 22, normalized size = 0.76 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \relax (x)\right )}}{c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c*x)

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giac [A]  time = 1.32, size = 35, normalized size = 1.21 \begin {gather*} -\frac {a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

-a*log(abs(-sqrt(c)*x + sqrt(c*x^2)))/sqrt(c) + sqrt(c*x^2)*b/c

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maple [A]  time = 0.00, size = 18, normalized size = 0.62 \begin {gather*} \frac {\left (a \ln \relax (x )+b x \right ) x}{\sqrt {c \,x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(c*x^2)^(1/2),x)

[Out]

1/(c*x^2)^(1/2)*x*(a*ln(x)+b*x)

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maxima [A]  time = 1.32, size = 20, normalized size = 0.69 \begin {gather*} \frac {a \log \relax (x)}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

a*log(x)/sqrt(c) + sqrt(c*x^2)*b/c

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mupad [B]  time = 0.51, size = 17, normalized size = 0.59 \begin {gather*} \frac {b\,\relax |x|+a\,\ln \left (c\,x\right )\,\mathrm {sign}\relax (x)}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(c*x^2)^(1/2),x)

[Out]

(b*abs(x) + a*log(c*x)*sign(x))/c^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\sqrt {c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(c*x**2)**(1/2),x)

[Out]

Integral((a + b*x)/sqrt(c*x**2), x)

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